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Unit: A1.7

Quadratic Equations

• Connect solving quadratic equations to finding zeros of quadratic functions (A-SSE.B.3a$^\star$, F-IF.C.8a$^\star$).
• Explore forms of quadratic equations that can be solved by seeing structure (A-REI.B.4b).
• Understand and be able to use the method of factoring to solve factorable quadratic equations (A-REI.A.1, A-REI.B.4b).
• Understand and be able to use the method of completing the square to solve quadratic equations, and derive the quadratic formula (A-SSE.B.3b$^\star$, A-REI.A.1, A-REI.B.4).
• Construct and solve quadratic equations by the most strategic method to solve problems in various contexts (A-CED.A.1$^\star$, A-REI.B.4b).
• Express a quadratic function in the appropriate form for a given purpose, including vertex form (A-SSE.B.3b$^\star$, F-IF.C.8a$^\star$).
• Solve problems using systems consisting of a linear and a quadratic equation in two variables (A-REI.C.7).
• Derive the equation of a parabola given the focus and a directrix parallel to one of the axes (G-GPE.A.2).

The story before this unit:

Students have just completed a study of quadratic functions where they explored tables, graphs, and expressions for quadratic functions in situations that can be modeled by them. The various forms of a quadratic expression were necessitated and explored.

The part of the story happening in this unit:

In this unit, students see quadratic equations arise naturally out of modeling problems involving quadratic functions. Any time you want to know the input to a quadratic function that produces a specified output, you have to solve a quadratic equation. Students understand solving quadratic equations as a process of reasoning and use the properties of operations to form equivalent quadratic expressions and equations. They convert between standard, vertex, and factored form by factoring, completing the square, and distributing. This unit does not treat factoring as a systematic method, but rather as an opportunistic method to be used when a factorization is readily available. The method of completing the square is a systematic method that works in all cases, and leads to the quadratic formula. Students should be able to solve quadratic equations by many different methods, making strategic choices of the best method for the situation at hand.

Students then adapt the method of completing the square to putting a quadratic function in vertex form. The unit ends with a section on deriving the equation for a parabola from its geometric definition.

The story after this unit:

Students will start to encounter situations where the roots are not real numbers, which will prepare them for a full investigation of complex numbers in the next unit.

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Sections

A1.7.0 Pre-unit diagnostic assessment

Summary

Diagnose students’ ability to

• recognize a solution to a quadratic equation (A-REI.A.1);
• perform operations on rational numbers (7.NS.A);
• work with irrational numbers, primarily by approximating on the number line (N-RN.B.3).

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A1.7.1 Motivate solving quadratic equations

Summary

• Generate an intellectual need to solve a quadratic equation.
• Understand the graphical method for solving equations approximately.

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A1.7.2 What do we already know about solving quadratic equations?

Summary

• Understand the Zero Product Property and use it to justify steps to solve a factorable quadratic equation (A-REI.A.1).
• Explore forms of quadratic equations that can be solved by seeing structure (A-SSE.A.2, A-REI.B.4b).
• Connect solving quadratic equations to finding zeros of quadratic functions (A-SSE.B.3A$^\star$, F-IF.C.8a$^\star$).

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A1.7.3 Solve quadratic equations by completing the square

Summary

Understand and be able to use the method of completing the square to solve quadratic equations (A-REI.A.1, A-REI.B.4).

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A1.7.4 Put quadratic functions in vertex form

Summary

Put quadratic functions in vertex form (A-SSE.B.3b$^\star$, F-IF.C.8a$^\star$).

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A1.7.5 Deriving the quadratic formula

Summary

Use completing the square to derive the quadratic formula (A-SSE.B.3b$^\star$, A-REI.B.4a).

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A1.7.6 Practice with all methods, picking the best one to use, and drawing connections between them

Summary

• Construct and solve quadratic equations by the most strategic method in various contexts (A-CED.A.1$^\star$, A-REI.B.4b).
• Express a quadratic function in the appropriate form for a given purpose (F-IF.C.8a$^\star$).

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A1.7.7 Connect algebra to geometry

Summary

• Solve problems using systems of a linear and a quadratic equation in two variables (A-REI.C.7).
• Derive equation of a parabola given the focus and a directrix parallel to one of the axes (G-GPE.A.2).

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A1.7.8 Culminating activity

Summary

Apply the tools learned in this unit to a real-world situation.

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A1.7.9 Summative assessment

Summary

Assess students’ ability to

• reason with algebraic properties to solve quadratic equations (A-REI.A.1);
• select the best method to solve a quadratic equation (factor, complete the square, quadratic formula) (A-REI.B.4);
• relate solutions of quadratic equations to graphs and interpret solutions in terms of a context (A-SSE.B.3a$^\star$, F-IF.C.8a$^\star$).

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